/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([1, 5, -5, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([13, 13, -w^2 + 3]) primes_array = [ [5, 5, w^3 - 4*w],\ [7, 7, w + 1],\ [7, 7, -w^3 + 4*w - 1],\ [13, 13, -w^2 + 3],\ [16, 2, 2],\ [17, 17, w^3 + w^2 - 4*w - 2],\ [17, 17, -w^3 + 5*w - 2],\ [19, 19, -w^2 - w + 4],\ [19, 19, -w^2 - w + 1],\ [29, 29, 2*w^3 - w^2 - 9*w + 5],\ [41, 41, -w^3 + w^2 + 5*w - 3],\ [43, 43, w^3 - w^2 - 4*w + 2],\ [43, 43, w^3 - 6*w],\ [47, 47, -w^3 - w^2 + 5*w],\ [49, 7, w^2 + 2*w - 2],\ [59, 59, 2*w^3 - 8*w + 3],\ [67, 67, w^2 - w - 4],\ [79, 79, w^3 - w^2 - 4*w + 1],\ [81, 3, -3],\ [97, 97, w^3 + w^2 - 5*w + 1],\ [107, 107, -2*w^3 + 3*w^2 + 10*w - 12],\ [109, 109, 2*w^3 - w^2 - 9*w + 3],\ [125, 5, -3*w^3 - w^2 + 12*w + 1],\ [127, 127, -2*w^3 + w^2 + 10*w - 8],\ [137, 137, -w^3 + w^2 + 3*w - 5],\ [139, 139, w^3 - w^2 - 5*w + 1],\ [149, 149, 2*w^2 - w - 6],\ [149, 149, 2*w^3 + w^2 - 9*w],\ [163, 163, w^3 - 5*w - 3],\ [163, 163, 2*w^3 - 7*w],\ [173, 173, -2*w^3 + 11*w - 3],\ [173, 173, -w^3 + w^2 + 4*w - 8],\ [179, 179, 2*w^3 - w^2 - 8*w + 5],\ [181, 181, 2*w^3 - w^2 - 8*w],\ [181, 181, 2*w^3 - 9*w - 3],\ [191, 191, 2*w^2 - 5],\ [191, 191, -2*w^2 - w + 10],\ [193, 193, w^3 + w^2 - 3*w - 4],\ [193, 193, -w^3 + 2*w^2 + 3*w - 3],\ [197, 197, -2*w^3 + 9*w - 4],\ [197, 197, w^2 - 7],\ [199, 199, 3*w^3 - 13*w + 1],\ [199, 199, w^3 + 2*w^2 - 5*w - 7],\ [227, 227, -3*w^3 + w^2 + 13*w - 4],\ [227, 227, -w^3 + 2*w^2 + 7*w - 6],\ [241, 241, -2*w^3 + 7*w - 1],\ [257, 257, -2*w^3 + w^2 + 7*w - 5],\ [257, 257, 2*w^3 - w^2 - 7*w + 2],\ [263, 263, -w^3 + 7*w - 3],\ [269, 269, -2*w^3 + 2*w^2 + 11*w - 8],\ [271, 271, w^2 + 3*w - 2],\ [271, 271, w^2 + 2*w - 7],\ [277, 277, -2*w^3 + 11*w - 2],\ [281, 281, 2*w^3 - 11*w + 1],\ [283, 283, 2*w^3 - 9*w + 5],\ [283, 283, -2*w^3 + w^2 + 11*w - 7],\ [289, 17, -w^3 + 2*w^2 + 4*w - 4],\ [293, 293, -2*w^3 - w^2 + 7*w + 3],\ [307, 307, -3*w^3 + 2*w^2 + 13*w - 9],\ [311, 311, -w^3 + w^2 + 3*w - 7],\ [313, 313, -w^3 + 2*w^2 + 3*w - 2],\ [313, 313, w^2 - w - 8],\ [317, 317, -2*w^3 - w^2 + 9*w - 1],\ [331, 331, w^3 + 2*w^2 - 5*w - 3],\ [337, 337, w^3 + 2*w^2 - 6*w - 4],\ [337, 337, w^3 - 2*w^2 - 6*w + 5],\ [347, 347, 2*w^3 + 2*w^2 - 9*w - 3],\ [347, 347, -2*w^3 + w^2 + 7*w - 4],\ [349, 349, w^2 - 2*w - 4],\ [353, 353, -w^3 + 2*w^2 + 2*w - 6],\ [359, 359, -w^3 + w^2 + 7*w - 6],\ [359, 359, -w^2 - w - 2],\ [361, 19, 2*w^3 - w^2 - 11*w],\ [367, 367, -2*w^3 + 2*w^2 + 8*w - 5],\ [373, 373, w^3 + w^2 - 2*w - 4],\ [379, 379, 2*w^3 - w^2 - 11*w + 4],\ [389, 389, 3*w^2 + w - 11],\ [397, 397, 3*w^3 - 13*w + 5],\ [401, 401, w - 5],\ [409, 409, 3*w^3 - 12*w + 4],\ [409, 409, 3*w^3 + 2*w^2 - 14*w - 4],\ [421, 421, 2*w^3 - w^2 - 10*w + 1],\ [421, 421, w^3 + 2*w^2 - 5*w - 4],\ [439, 439, -2*w^3 + 10*w - 5],\ [443, 443, -2*w^3 + 12*w - 5],\ [443, 443, -4*w^3 + w^2 + 18*w - 9],\ [443, 443, -2*w^3 + w^2 + 7*w - 9],\ [443, 443, 3*w^3 + 2*w^2 - 14*w - 5],\ [449, 449, w^3 + 2*w^2 - 2*w - 6],\ [449, 449, -2*w^3 + w^2 + 11*w - 5],\ [461, 461, 2*w^3 + 2*w^2 - 9*w - 4],\ [467, 467, -w^3 + w^2 + 7*w - 5],\ [487, 487, 2*w^2 + 4*w - 7],\ [491, 491, -w^3 + w^2 + 7*w - 4],\ [503, 503, 2*w^3 + w^2 - 9*w + 2],\ [521, 521, w^2 - 3*w - 3],\ [523, 523, -w^3 + 2*w^2 + 5*w - 4],\ [523, 523, -w^3 + 8*w - 5],\ [563, 563, w^3 + 3*w^2 - w - 8],\ [563, 563, w^3 - 2*w^2 - 8*w + 4],\ [563, 563, -w^3 + w^2 + 2*w - 5],\ [563, 563, -3*w^3 - w^2 + 10*w - 2],\ [577, 577, 3*w^3 - 16*w],\ [587, 587, w^3 + 2*w^2 - 6],\ [587, 587, -w^3 + 4*w - 6],\ [601, 601, -w - 5],\ [607, 607, w^3 - 2*w^2 - 4*w + 1],\ [613, 613, 4*w^3 - 17*w],\ [617, 617, -3*w^3 + w^2 + 14*w - 2],\ [617, 617, -4*w^3 + 17*w - 4],\ [619, 619, -3*w^3 + 12*w - 2],\ [643, 643, 3*w^3 - w^2 - 12*w + 3],\ [643, 643, -2*w^3 - w^2 + 11*w - 2],\ [647, 647, 2*w^3 + w^2 - 9*w + 3],\ [653, 653, w^3 + w^2 - 7*w - 4],\ [661, 661, -w^3 + 4*w^2 + 6*w - 15],\ [661, 661, 3*w^3 - 2*w^2 - 12*w + 10],\ [677, 677, -3*w^2 - 2*w + 12],\ [691, 691, -2*w^3 + 3*w^2 + 11*w - 10],\ [691, 691, -3*w^3 + 16*w - 3],\ [701, 701, 3*w^3 - 13*w + 6],\ [719, 719, -3*w^2 + w + 10],\ [727, 727, w^3 - 3*w^2 - 4*w + 6],\ [743, 743, -w^3 + 2*w^2 + 5*w - 3],\ [761, 761, -w^3 + 3*w^2 + 3*w - 12],\ [761, 761, 3*w^3 - 2*w^2 - 14*w + 7],\ [769, 769, -3*w^3 + 3*w^2 + 16*w - 13],\ [769, 769, w^3 + 2*w^2 - 7*w - 6],\ [773, 773, w^3 - 8*w + 4],\ [773, 773, 4*w^2 + w - 16],\ [787, 787, w^2 + 4*w - 2],\ [797, 797, -2*w^3 + 3*w^2 + 10*w - 10],\ [797, 797, -w^3 - w^2 + 4*w - 4],\ [797, 797, w^3 + 2*w^2 - 3*w - 9],\ [797, 797, -3*w^3 + 3*w^2 + 11*w - 10],\ [821, 821, w^2 + w - 9],\ [827, 827, w^3 - 8*w + 1],\ [827, 827, -2*w^3 + w^2 + 9*w - 11],\ [827, 827, 3*w^3 - 14*w + 5],\ [827, 827, 2*w^3 + w^2 - 7*w - 5],\ [829, 829, -2*w^3 - 2*w^2 + 9*w - 1],\ [839, 839, -3*w^3 + 15*w - 5],\ [853, 853, -4*w^3 + 15*w - 8],\ [853, 853, -w^3 + 2*w^2 + 6*w - 3],\ [859, 859, -w^3 + w^2 + w - 4],\ [859, 859, w^3 + w^2 - w - 5],\ [881, 881, -w^3 + w^2 + 2*w - 6],\ [881, 881, w^3 + w^2 - 2*w - 6],\ [883, 883, w^3 + 3*w^2 - 3*w - 6],\ [887, 887, w^3 + 3*w^2 - 3*w - 8],\ [919, 919, 2*w^3 - w^2 - 13*w],\ [929, 929, -w^3 - 3*w^2 + 2*w + 10],\ [929, 929, w^3 + w^2 - 5*w - 8],\ [941, 941, w^3 + 2*w^2 - 3*w - 10],\ [953, 953, -w^3 + 5*w - 7],\ [953, 953, -2*w^3 + 3*w^2 + 9*w - 9],\ [953, 953, -2*w^3 - w^2 + 10*w - 3],\ [953, 953, -3*w^3 + 11*w - 1],\ [961, 31, -w^3 + 3*w^2 + 2*w - 10],\ [961, 31, -3*w^2 + 2*w + 7],\ [967, 967, w^3 + 3*w^2 - 3*w - 7],\ [967, 967, 4*w^3 + w^2 - 20*w + 1],\ [983, 983, 2*w^3 - w^2 - 12*w + 8],\ [983, 983, 2*w^3 + w^2 - 6*w - 4],\ [991, 991, 2*w^3 - 12*w + 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 7*x^3 + 13*x^2 + x - 13 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -2*e^3 + 9*e^2 - 5*e - 8, -e^2 + 4*e - 1, -1, -e^3 + 5*e^2 - e - 11, e^3 - 3*e^2 - 2*e + 6, -e + 3, 3*e^3 - 15*e^2 + 11*e + 14, e^3 - 4*e^2 + e + 5, 3*e^3 - 13*e^2 + 4*e + 22, -e^3 + 4*e^2 - 5*e + 4, -3*e^3 + 13*e^2 - 5*e - 12, e^3 - 6*e^2 + 8*e + 3, -e^3 + 8*e^2 - 10*e - 13, -4*e^3 + 16*e^2 - 3*e - 22, 2*e^3 - 9*e^2 + e + 16, 7*e^3 - 30*e^2 + 10*e + 38, 5*e^3 - 23*e^2 + 11*e + 28, 5*e^3 - 25*e^2 + 14*e + 35, e^3 - 3*e^2 - 5*e + 6, e^3 - 8*e^2 + 14*e + 5, -e^3 + 5*e^2 - 2*e - 17, 5*e^3 - 28*e^2 + 21*e + 40, 4*e^3 - 23*e^2 + 16*e + 38, 3*e^3 - 11*e^2 + 3*e + 15, -e^3 + 9*e^2 - 11*e - 21, -4*e^3 + 23*e^2 - 22*e - 18, -e^3 + 2*e^2 + 2*e + 2, -5*e^3 + 28*e^2 - 25*e - 34, 2*e^3 - 12*e^2 + 13*e + 23, -6*e^3 + 26*e^2 - 10*e - 29, 3*e^3 - 6*e^2 - 13*e + 9, -e^3 - e^2 + 5*e + 24, 7*e^3 - 31*e^2 + 10*e + 41, -e^3 + 2*e^2 - 5*e + 13, -2*e^3 + 10*e^2 - 8*e - 3, -7*e^3 + 31*e^2 - 17*e - 38, 9*e^3 - 33*e^2 - 3*e + 51, -e^3 + 2*e^2 + 9, 8*e^3 - 38*e^2 + 20*e + 61, -11*e^3 + 51*e^2 - 32*e - 50, -e^2 + 9*e, e^3 - 11*e^2 + 24*e + 5, 6*e^3 - 30*e^2 + 24*e + 43, 7*e^3 - 32*e^2 + 22*e + 16, 2*e^3 - 3*e^2 - 5*e - 3, -14*e^3 + 62*e^2 - 32*e - 67, -3*e^3 + 16*e^2 - 10*e - 30, -5*e^3 + 13*e^2 + 6*e + 7, 14*e^3 - 63*e^2 + 27*e + 87, -2*e^3 + 8*e^2 - 6*e - 3, -7*e^3 + 38*e^2 - 31*e - 49, -5*e^3 + 20*e^2 + e - 33, 3*e^3 - 25*e^2 + 38*e + 19, 11*e^3 - 51*e^2 + 27*e + 51, -3*e^3 + 23*e^2 - 33*e - 20, 3*e^3 - 9*e^2 - 9*e + 20, 3*e^3 - 17*e^2 + 20*e + 5, -15*e^3 + 69*e^2 - 40*e - 75, 2*e^3 - 12*e^2 + 3*e + 42, -9*e^3 + 38*e^2 - 19*e - 32, 6*e^3 - 24*e^2 + 2*e + 45, 3*e^2 - 12*e + 9, -6*e^3 + 33*e^2 - 27*e - 42, -4*e^3 + 12*e^2 + 7*e - 14, -8*e^2 + 18*e + 7, 17*e^3 - 77*e^2 + 30*e + 108, 10*e - 13, -9*e^3 + 50*e^2 - 44*e - 54, -3*e^3 + 15*e^2 - 4*e - 39, 11*e^3 - 51*e^2 + 20*e + 71, -e^3 + 3*e^2 + 13*e - 23, 5*e^3 - 28*e^2 + 23*e + 40, -14*e^3 + 64*e^2 - 28*e - 88, e^3 + 6*e^2 - 21*e - 23, 8*e^3 - 30*e^2 + 2*e + 49, -5*e^3 + 29*e^2 - 15*e - 65, -14*e^3 + 64*e^2 - 28*e - 86, -12*e^3 + 53*e^2 - 21*e - 76, 7*e^3 - 33*e^2 + 22*e + 24, -e^3 + 6*e^2 - 5*e + 6, -3*e^3 + 7*e^2 + 8*e, -3*e^3 + 9*e^2 + 4*e + 8, 4*e^3 - 25*e^2 + 32*e + 25, e^3 - 3*e^2 - 3*e - 10, 2*e^3 - 14*e^2 + 9*e + 21, 10*e^3 - 42*e^2 + 14*e + 55, -15*e^3 + 62*e^2 - 25*e - 57, 11*e^3 - 52*e^2 + 23*e + 74, -9*e^3 + 39*e^2 - 7*e - 60, 11*e^3 - 44*e^2 + 15*e + 42, 6*e^3 - 27*e^2 + 11*e + 50, 4*e^3 - 27*e^2 + 35*e + 8, -3*e^3 + 17*e^2 - 2*e - 34, -10*e^3 + 39*e^2 - 12*e - 40, -9*e^3 + 51*e^2 - 56*e - 49, 5*e^3 - 31*e^2 + 37*e + 30, -2*e^3 + 15*e^2 - 21*e - 25, -3*e^3 + 17*e^2 - 24*e - 19, -15*e^3 + 67*e^2 - 25*e - 96, -6*e^3 + 31*e^2 - 27*e - 25, -3*e^3 + 8*e^2 + 22*e - 34, 11*e^3 - 47*e^2 + 30*e + 36, -10*e^3 + 42*e^2 - 17*e - 41, -7*e^3 + 23*e^2 + 6*e - 24, -22*e^3 + 107*e^2 - 70*e - 115, -14*e^3 + 71*e^2 - 45*e - 75, 23*e^3 - 103*e^2 + 42*e + 138, 7*e^3 - 20*e^2 - 16*e + 13, -19*e^3 + 94*e^2 - 55*e - 124, -5*e^3 + 32*e^2 - 30*e - 49, -18*e^3 + 86*e^2 - 39*e - 122, -8*e^3 + 30*e^2 + e - 30, -8*e^3 + 33*e^2 - 8*e - 59, -18*e^3 + 73*e^2 - 18*e - 92, 15*e^3 - 75*e^2 + 51*e + 75, -e^3 + 8*e^2 - 24*e + 20, 17*e^3 - 72*e^2 + 32*e + 57, -3*e^3 + 13*e^2 - 22*e - 4, 14*e^3 - 61*e^2 + 21*e + 109, -4*e^3 + 14*e^2 + 10*e - 27, -4*e^3 + 9*e^2 - e + 27, 17*e^3 - 71*e^2 + 9*e + 118, 3*e^3 - 12*e^2 + 5*e + 15, -e^3 + 4*e^2 - 13*e + 30, 14*e^3 - 72*e^2 + 50*e + 79, 2*e^3 - 3*e^2 - 9*e + 3, -23*e^3 + 109*e^2 - 70*e - 125, -14*e^3 + 67*e^2 - 33*e - 117, 5*e^3 - 14*e^2 - 11*e + 12, -17*e^3 + 80*e^2 - 47*e - 77, 7*e^3 - 21*e^2 - 26*e + 51, 8*e^3 - 22*e^2 - 9*e - 7, 2*e^3 - 16*e^2 + 37*e - 8, -8*e^3 + 36*e^2 - 16*e - 88, -7*e^3 + 37*e^2 - 22*e - 84, -14*e^3 + 71*e^2 - 58*e - 71, 9*e^3 - 47*e^2 + 43*e + 66, -11*e^3 + 46*e^2 - 13*e - 40, 10*e^3 - 39*e^2 + 21*e + 16, -9*e^3 + 45*e^2 - 21*e - 71, -12*e^3 + 62*e^2 - 51*e - 69, -15*e^3 + 72*e^2 - 47*e - 57, -4*e^3 + 23*e^2 - 40*e - 11, -7*e^3 + 21*e^2 - e + 19, -4*e^3 + 25*e^2 - 45*e - 19, 11*e^3 - 53*e^2 + 49*e + 36, 3*e^3 - 12*e^2 + 5*e - 1, -4*e^3 + 14*e^2 + 3*e - 34, -22*e^3 + 106*e^2 - 54*e - 148, 9*e^3 - 45*e^2 + 32*e + 26, 4*e^3 - 7*e^2 - 39*e + 20, 6*e^3 - 23*e^2 + 5*e - 9, -15*e^3 + 78*e^2 - 64*e - 69, -7*e^3 + 15*e^2 + 31*e - 31, 10*e^3 - 34*e^2 - 3*e + 36, 2*e^3 - 5*e^2 - 4*e + 12, -7*e^3 + 26*e^2 - 4*e - 35, 10*e^3 - 35*e^2 + 2*e + 36, -6*e^3 + 38*e^2 - 42*e - 34, 23*e^3 - 112*e^2 + 80*e + 112, 2*e^3 - 14*e^2 + 38*e - 14, -17*e^3 + 73*e^2 - 28*e - 100, 16*e^3 - 78*e^2 + 46*e + 81, 11*e^3 - 42*e^2 - 8*e + 71] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([13, 13, -w^2 + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]